Algebraic & Graphical interpretation of Simultaneous Equation
Let’s take a pair of linear equation:
……………..(1) 
……………..(2)
With the help of coefficient of equations (1) & (2) we can easily determine if the solution of the simultaneous equations will be Unique Solution, No Solution or Infinite Solution.
Let’s concentrate on the coefficient of the equation now and calculate the following :
i.) 
ii.) 
iii.) 
and compare the ratios.
Unique solution : Algebraic Interpretation
If 
Number of solutions: Unique solution
Graphical Representation : Two intersecting lines
Type of Equation : Independent
Example :
2x – 3y = 1 ……………(1)
3x – 4y = 1 ……………(2)
In this case 
So, . Hence Unique solution.
Unique solution : Graphical Interpretation Let's look at the graph of the two equations 2x – 3y = 1 and 3x – 4y = 1 on the same plot.
No solution : Algebraic Interpretation
If 
Number of solutions: No solution
Graphical Representation : Two parallel lines
Type of Equation : Inconsistent
Example :
2x – 3y = 4 ……………(1)
4x – 6y = 12 ……………(2)
In this case 
So, . Hence No solution.
No solution : Graphical Interpretation Let's look at the graph of the two equations 2x – 3y = 4 and 4x – 6y = 12 on the same plot.
Infinite Number of solution : Algebraic Interpretation
If 
Number of solutions: Infinite number of solution
Graphical Representation : Same Line
Type of Equation : Dependent
Example :
2x – 3y = 4 ……………(1)
4x – 6y = 8 ……………(2)
In this case
So, . Hence, Infinite number of solution.
Infinite Number of solution : Graphical Interpretation Let's look at the graph of the two equations 2x – 3y = 4 and 4x – 6y = 8 on the same plot.
So, to solve a system of simultaneous equations, number of independent equations must be equal to the number of variables.
